The line `l x+m y+n=0` is a normal to the ellipse `(x^2)/(a^2)+(y^2)/(b^2)=1` . then prove that `(a^2)/(l^2)+(b^2)/(m^2)=((a^2-b^2)^2)/(n^2)`

For the ellipse (x^(2))/(a^(2))+(y^(2))/(b^(2)) =1 and (x^(2))/(b^(2))+(y^(2))/(a^(2)) =1

Find the maximum area of the ellipse (x^2)/(a^2)+(y^2)/(b^2)=1 which touches the line y=3x+2.

Find the equation of the normal to the ellipse (x^2)/(a^2)+(y^2)/(b^2)=1 at the positive end of the latus rectum.

A normal to the hyperbola (x^2)/4-(y^2)/1=1 has equal intercepts on the positive x- and y-axis. If this normal touches the ellipse (x^2)/(a^2)+(y^2)/(b^2)=1 , then a^2+b^2 is equal to (a)5 (b) 25 (c) 16 (d) none of these

Prove that the chords of constant of perpendicular tangents to the ellipse (x^(2))/(a^(2))+(y^(2))/(b^(2))=1 touch another fixed ellipse (x^(2))/(a^(4))+(y^(2))/(b^(4))=(1)/((a^(2)+b^(2)))

A tangent to the hyperbola (x^2)/(a^2)-(y^2)/(b^2)=1 cuts the ellipse (x^2)/(a^2)+(y^2)/(b^2)=1 at Pa n dQ . Show that the locus of the midpoint of P Q is ((x^2)/(a^2)+(y^2)/(b^2))^2=(x^2)/(a^2)-(y^2)/(b^2)dot

Find the equations of the tangent and normal to the hyperbola (x^(2))/(a^(2))-(y^(2))/(b^(2))=1 at the point (x_(0), y_(0)).

If any line perpendicular to the transverse axis cuts the hyperbola (x^2)/(a^2)-(y^2)/(b^2)=1 and the conjugate hyperbola (x^2)/(a^2)-(y^2)/(b^2)=-1 at points Pa n dQ , respectively, then prove that normal at Pa n dQ meet on the x-axis.

(1) Draw the rough sketch of the ellipse (x^(2))/(a^(2)) + (y^(2))/(b^(2)) = 1 . Find the area enclosed by the ellipse (x^(2))/(a^(2)) + (y^(2))/(b^(2)) = 1 .